LAUSR

Abstract Let 0 ≤ α ≤ 1 and ϕ be an integer function defined on N ∖ { 0 } satisfying 1 ≤ ϕ ( n ) ≤ n . Define the level set E R ϕ ( α ) = { x ∈ [ 0 , 1 ] : lim n → ∞ ⁡ A n , ϕ ( n ) ( x ) = α } , where A n , ϕ ( n ) ( x ) is the ( n , ϕ ( n ) ) -Erdos–Renyi average of x ∈ [ 0 , 1 ] . In this paper, we will give descriptions for the Hausdorff dimension of E R ϕ ( α ) under the assumption ϕ ( n ) → ∞ as n → ∞ , which complement simultaneously an early classic result of Besicovitch and the new strong law of large number established by P. Erdos and A. Renyi. Moreover, for the case ϕ ( n ) = M ultimately, where M ≥ 1 is an integer, the Hausdorff dimension of E R ϕ ( α ) is also determined by us in the last section.